Some Related Orderings

Abstract

In Sect. 6.4, we encountered some alternative inequality curves and the corresponding orderings that they can be used to define (they were Bonferroni, Leimkuhler, and Zenga-I and II). In this chapter we consider certain other partial orders defined on $\mathcal {L}$ (the class of non-negative random variables with positive finite expectations) that are closely related to the Lorenz ordering and are sometimes useful to aid in the determination of whether or not random variables are Lorenz ordered. The first, ∗-ordering, is often easier to deal with than Lorenz ordering, and is a prime example of an ordering that can sometimes be used to verify Lorenz ordering which it implies. The other group of partial orderings to be discussed are those known as stochastic dominance of degree k, k = 1, 2, … Degree 1 is just stochastic ordering. Degree 2 is intimately related to the Lorenz order, but distinct. Higher degree stochastic orders are most frequently encountered in economic contexts. The treatment provided here is brief.